An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 3. 4. Fractions. 5
is impossible if the divisor be zero. For if the dividend be not zero,
there is no number, which when multiplied by zero, gives a product
equal to it, since a product one of whose factors is zero vanishes;
but if the dividend is likewise zero, the value of the quotient is completely indeterminate; no calculation can be performed with such a
completely indeterminate number.
Integer and fractional numbers are both embraced under the expression Rational Numbers. Applying the first four rules of Arithmetic to them we always reproduce rational numbers. The series of
rational numbers is unlimited: between any two, as a continued subdivision shows, we can always insert as many more rational numbers as
we please, or to state the same thing in other words, a series of
rational numbers which is arrived at by any finite number of subdivisions never forms a continuum.